Inferring Structure via Duality for Photonic Inverse Design

Led by a result derived from Sion's minimax theorem concerning constraint violation in quadratically constrained quadratic programs (QCQPs) with at least one constraint bounding the possible solution magnitude, we propose a heuristic scheme for photonic inverse design unifying core ideas from adjoint optimization and convex relaxation bounds. Specifically, through a series of alterations to the underlying constraints and objective, the QCQP associated with a given design problem is gradually transformed so that it becomes strongly dual. Once equivalence between primal and dual programs is achieved, a material geometry is inferred from the solution of the modified QCQP. This inferred structure, due to the complementary relationship between the dual and primal programs, encodes overarching features of the optimization landscape that are otherwise difficult to synthesize, and provides a means of initializing secondary optimization methods informed by the global problem context. An exploratory implementation of the framework, presented in a partner manuscript, is found to achieve dramatic improvements for the exemplary photonic design task of enhancing the amount of power extracted from a dipole source near the boundary of a structured material region -- roughly an order of magnitude compared to randomly initialized adjoint-based topology optimization for areas surpassing $10~\lambda^{2}$.